Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395190 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14886 |
| prime radical | #23434 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,14)(4,18)(5,12)(6,11)(7,13)(8,16)(9,19)(10,20)(15,23)(17,28)(21,25)(22,27)(24,30)(26,31)(29,32);(1,3,5,16,7,17,6,21)(2,8,11,14,13,25,12,28)(4,15,19,29,20,31,22,30)(9,26,18,32,27,23,10,24);(1,4)(2,9)(3,15)(5,19)(6,22)(7,20)(8,23)(10,12)(11,18)(13,27)(14,24)(16,29)(17,31)(21,30)(25,26)(28,32);(1,4)(2,10,13,18)(3,15)(5,19)(6,22)(7,20)(8,24,25,32)(9,12,27,11)(14,26,28,23)(16,29)(17,31)(21,30)Polycyclic presentation:
2^9|b^2=e,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280193, #72339020, #72339028, … |
| maximal normal subgroup of | #72280193, #72339020, #72339028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311323);